Reflections on Tiles (in Self-Assembly)

We define the Reflexive Tile Assembly Model (RTAM), which is obtained from the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across their horizontal and/or vertical axes. We show that the class of directed temperature-1 RTAM systems is not computationally universal, which is conjectured but unproven for the aTAM, and like the aTAM, the RTAM is computationally universal at temperature 2. We then show that at temperature 1, when starting from a single tile seed, the RTAM is capable of assembling n x n squares for n odd using only n tile types, but incapable of assembling n x n squares for n even. Moreover, we show that n is a lower bound on the number of tile types needed to assemble n x n squares for n odd in the temperature-1 RTAM. The conjectured lower bound for temperature-1 aTAM systems is 2n-1. Finally, we give preliminary results toward the classification of which finite connected shapes in Z^2 can be assembled (strictly or weakly) by a singly seeded (i.e. seed of size 1) RTAM system, including a complete classification of which finite connected shapes be strictly assembled by a "mismatch-free" singly seeded RTAM system.Comment: New results which classify the types of shapes which can self-assemble in the RTAM have been adde

Force induced triple point for interacting polymers

We show the existence of a force induced triple point in an interacting polymer problem that allows two zero-force thermal phase transitions. The phase diagrams for three different models of mutually attracting but self avoiding polymers are presented. One of these models has an intermediate phase and it shows a triple point but not the others. A general phase diagram with multicritical points in an extended parameter space is also discussed.Comment: 4 pages, 8 figures, revtex

Constructive Dimension and Turing Degrees

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim_H(S) and constructive packing dimension dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) / dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0, then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness extractor* that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) = dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems, 45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to insufficient care with the choice of delta. This version modifies that proof to fix the error

Dynamics and Transport in Random Antiferromagnetic Spin Chains

We present the first results on the low-frequency dynamical and transport properties of random antiferromagnetic spin chains at low temperature ($T$). We obtain the momentum and frequency dependent dynamic structure factor in the Random Singlet (RS) phases of both spin-1/2 and spin-1 chains, as well as in the Random Dimer phase of spin-1/2 chains. We also show that the RS phases are unusual `spin-metals' with divergent low-frequency conductivity at T=0, and follow the spin conductivity through `metal-insulator' transitions tuned by the strength of dimerization or Ising anisotropy in the spin-1/2 case, and by the strength of disorder in the spin-1 case.Comment: 4 pages (two-column format). Presentation substantially revised to accomodate new result

The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square

In this paper we define the Dupled abstract Tile Assembly Model (DaTAM), which is a slight extension to the abstract Tile Assembly Model (aTAM) that allows for not only the standard square tiles, but also "duple" tiles which are rectangles pre-formed by the joining of two square tiles. We show that the addition of duples allows for powerful behaviors of self-assembling systems at temperature 1, meaning systems which exclude the requirement of cooperative binding by tiles (i.e., the requirement that a tile must be able to bind to at least 2 tiles in an existing assembly if it is to attach). Cooperative binding is conjectured to be required in the standard aTAM for Turing universal computation and the efficient self-assembly of shapes, but we show that in the DaTAM these behaviors can in fact be exhibited at temperature 1. We then show that the DaTAM doesn't provide asymptotic improvements over the aTAM in its ability to efficiently build thin rectangles. Finally, we present a series of results which prove that the temperature-2 aTAM and temperature-1 DaTAM have mutually exclusive powers. That is, each is able to self-assemble shapes that the other can't, and each has systems which cannot be simulated by the other. Beyond being of purely theoretical interest, these results have practical motivation as duples have already proven to be useful in laboratory implementations of DNA-based tiles

Systems with Multiplicative Noise: Critical Behavior from KPZ Equation and Numerics

We show that certain critical exponents of systems with multiplicative noise can be obtained from exponents of the KPZ equation. Numerical simulations in 1d confirm this prediction, and yield other exponents of the multiplicative noise problem. The numerics also verify an earlier prediction of the divergence of the susceptibility over an entire range of control parameter values, and show that the exponent governing the divergence in this range varies continuously with control parameter.Comment: Four pages (In Revtex format) with 4 figures (in Postcript

Interconnected Self-Propagating Photopolymer Waveguides: An Alternative to Stereolithography for Rapid Formation of Lattice-Based Open-Cellular Materials

Recently, a new technique has been developed to create unique open-cellular materials with micro-scale truss, or lattice features ranging from tens to hundreds of microns. These materials are formed from a three-dimensional, interconnected array of self-propagating photopolymer waveguides. By utilizing this self-propagating effect, three-dimensional open-cellular polymer materials can be formed in seconds. In addition, intrinsic to the process is the ability to control specific micro-lattice parameters which ultimately affect the bulk material properties. Unlike stereolithography, this new fabrication technique is rapid (~ minutes to form an entire part) and relies on a single two-dimensional exposure surface to form three-dimensional structures (thickness > 25 mm possible). This combination of speed and planar scalability opens the possibility for large-scale mass manufacturing. The utility of these new materials range from lightweight energy absorbing structures to thermal management materials to bio-scaffolds.Mechanical Engineerin

Empirical approaches for the investigation of toxicant-induced loss of tolerance.

It has been hypothesized that sensitivity to low-level chemical exposures develops in two steps: initiation by an acute or chronic chemical exposure, followed by triggering of symptoms by low levels of previously tolerated chemical inhalants, foods, or drugs. The Working Group on Toxicant-induced Loss of Tolerance has formulated a series of research questions to test this hypothesis: Do some individuals experience sensitivity to chemicals at levels of exposure unexplained by classical toxicological thresholds and dose-response relationships, and outside normally expected variation in the population? Do chemically sensitive subjects exhibit masking that may interfere with the reproducibility of their responses to chemical challenges? Does chemical sensitivity develop because of acute, intermittent, or continuous exposure to certain substances? If so, what substances are most likely to initiate this process? An experimental approach for testing directly the relationship between patients' reported symptoms and specific exposures was outlined in response to the first question, which was felt to be a key question. Double-blind, placebo-controlled challenges performed in an environmentally controlled hospital facility (environmental medical unit) coupled with rigorous documentation of both objective and subjective responses are necessary to answer this question and to help elucidate the nature and origins of chemical sensitivity

Density Matrix Renormalization Group Study of the S=1/2 Anisotropic Antiferromagnetic Heisenberg Chains with Quasiperiodic Exchange Modulation

The low energy behavior of the S=1/2 antiferromagnetic XY-like XXZ chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. It is found that the energy gap of the chain with length N scales as $\exp (-cN^{\omega})$ with nonuniversal exponent $\omega$ if the Ising component of the exhange coupling is antiferromagnetic. This behavior is expected to be the characteristic feature of the quantum spin chains with relevant aperiodicity. This is in contrast to the XY chain for which the precious mean exchange modulation is marginal and the gap scales as $N^{-z}$. On the contrary, it is also verified that the energy gap scales as $N^{-1}$ if the Ising component of the exhange coupling is ferromagnetic. Our results are not only consistent with the recent bosonization analysis of Vidal, Mouhanna and Giamarchi but also clarify the nature of the strong coupling regime which is inaccesssible by the bosonization approach.Comment: 8 pages, 15 figures, 1 table; Proceedings of the workshop 'Frontiers in Magnetism', Kyoto, Oct. 199